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    Real Analysis : Proof using Summation Integrals

    For numbers a1,....,an, define p(x) = a1x +a2x^2+....+anx^n for all x. Suppose that: (a1)/2 + (a2)/3 +....+ (an)/(n+1) = 0 Prove that there is some point x in the interval (0,1) such that p(x) = 0

    Real Analysis : Integrability

    Prove that if f : [a,b] ----> R is a bounded function that is continuous at all but finitely many points, then f is integrable over [a,b].

    Radius of Convergence

    The problem is to determine the radius of convergence of the Taylor Series for each of the functions below centered at x. We are to explain our conclusion in each case. I would like to see how to work each problem (including what the Taylor Series is) and what the explanation is. a) centered at and NOTE: I know the

    Taylor polynomial approximation

    Suppose that the function F:R->R has derivatives of all orders and that: F"(x) - F'(x) - F(x) = 0 for all x F(0)=1 and F'(0)=1 Find a recursive formula for the coefficients of the nth Taylor polynomial for F:R->R at x=0. Show that the Taylor expansion converges at every point.

    Infinite Series of Real Numbers (Absolute Convergence)

    Please see the attached file for the fully formatted problem. Define ak recursively by a1 = 1 and ak = (−1)k  1 + k sin  1 k  −1 ak−1, k > 1. Prove that P 1k =1 ak converges absolutely. Since this problem is an analysis problem, please be sure to be rigorous.

    Functions : Convergence and Limits

    Please see the attached file for the fully formatted problems. Let f be a real function defined by . 1) Evaluate f'(x), f''(x), f(0). Show that f has exactly two roots and , with . Find an interval of two consecutive real numbers within which the roots must lie. From now on, let us denote and these two (closed) in

    Power and Taylor series

    Interval of Convergence of a power series a. Consider the Power series sum of series from n=1 to infinity of FnX^n. Use the ratio test to determine the open interval on which the pwr series converges. b. Show that the Taylor series of the Fcn f(x) = x/(1-x-x^2) about x=0 is given by: x/(1-x-x^2) = sum of series at

    Real Analysis : Young's Inequality

    Note: * = infinite Suppose that the function f:[0,*)->R is continuous and strictly increasing, with f(0) = 0 and f([0,*)) = [0,*). Then define F(x) = the integral from 0 to x of f and G(x) = the integral from 0 to x of f^-1 for all x>=0 (a) Prove Young's Inequality: ab <= F(a) + G(b) for all a >= 0 and b >= 0 (b) N

    Real Analysis: Geometric Interpretation in Terms of Areas

    Note: * = infinite Suppose that the function f:[0,*) -> R is continuous and strictly increasing, and that f:(0,*) -> R is differentiable. Moreover, assume f(0) = 0. Consider the formula: the integral from 0 to x of f + the integral from 0 to f(x) of f^-1 =xf(x) for all x>= 0. How can I provide a geometric interpretation

    Real Analysis: Criteria for Integrability

    Suppose the continuous function f:[a,b]->R has the property that: The integral from c to d f<=0 whenever a<=c<d<=b Prove that f(x)<=0 for all x in [a,b]. Is this true if we require only integrability of the function?

    Real Analysis of Criteria for Integrability

    Suppose that the function f:[a,b]->R is integrable and there is a postive number m such that f(x) >= m for all x in [a,b]. Show that the reciprocal function 1/f:[a,b]->R is integrable by proving that for each partition P of the interval [a,b], U(1/f,P) - L(1/f,P) <= 1/m^2[U(f,P) - L(f,P)]

    Real Analysis: Criteria for Integrability

    Define f(x) = x^2 for all x in [0,1]. For each natural number n, compute L(f,Pn) and U(f,Pn), where Pn is the regular partition of [0,1] into n subintervals.Then use the Integrability Criterion to show that the function f:[0,1]->R is integrable.

    Real Analysis - Riemann Integrability

    Please see the attached file for the fully formatted problems. Prove that if f is integrable on [0, 1], then lim n !1 Z 1 0 x n f(x)dx = 0 Since this problem is an analysis problem, please be sure to be rigorous. It falls under the chapter on Integrability on R , where they define partition, refinement of a partition,

    Real Analysis: Riemann Integrability

    Please see the attached file for the fully formatted problem. Let E = { 1/n : n 2 N } . Prove that the function f(x) = ? 1 x 2 E 0 otherwise is integrable on [0,1]. What is the value of R 1 0 f(x)dx? Since this problem is an analysis problem, please be sure to be rigorous. It falls under the chapter on Integrabili

    MacLaurin Series And Laplace Transforms : Absolute Convergence

    Find MacLaurin Series for the given function f. Use the linearity of the Laplace Transform to obtain a series representation L(f)=F(s) Determine 5 values for which the series converges absolutley (and uniformly). Also show the Laplace transform exists, i.e. that it has exponential order alpha. Here are the functions. A) f

    Real Analysis : Derivatives

    If I say that the function f:R->R has two derivatives, with f(0) = f'(0) = 0 and the absolute value of f"(x) is less than or equal to one, if the absolute value of x is less than or equal to 1. How can I prove that: f(x) <= 1/2 if x <= 1

    Real Analysis : Proof of a Constant

    Please see the attached file for the fully formatted problem. Let > 0. Prove that log x  x for x large. Prove that there exists a constant C such that log x  C x for all x 2 [1, 1 ), C ! 1 as ! 0+, and C ! 0 as ! 1 Please justify all steps and be rigorous because it is an analysis problem. (Note: The probl

    Real Analysis: Mean Value Theorem

    Let r e-1/x2 i(x) = i 0 x740 x = 0 Show that the nth derivative of 1(x) exists for all n E N. Please justify all steps and be rigorous because it is an analysis problem. (Note: The problem falls under the chapter on Differentiability on IR in the section entitled The Mean Value Theorem.)

    The inverse cosine function has domain

    The inverse cosine function has domain [-1,1]and range [0, pi]. Prove that (cos^-1)'(x) = -1/ sqrt(1-x^2). This needs to be proved from a real analysis point of view not a calculus.

    Composition Series

    Write a composition series for the rotation group of the cube and show that it is indeed a composition series.

    Series Convergence: Two-sided Estimator

    Please see the attached file for the fully formatted problems. A simple technique to estimate values of the finite sums of reciprocals to natural numbers raised to positive power and to define if corresponding infinite series converge. Deduce two-sided estimator for the sums of the positive powers (p) of reciprocals to nat

    Limits of Functions Using Epsilon

    Evaluate the following limits using the epsilon - delta definition and the limit theorems: a) lim {x -> 0} (x^2 + cos x)/(2 - tan x) b) lim {x -> sqrt(pi)} ((pi - x^2)^(1/3))/(x + pi)

    Functions: Limit Definitions

    Using the definition of a limit (rather than the limit theorems) prove that lim {x -> a+} f(x) exists and find the limit in each of the following cases a) f(x) = x/|x|, a = 0. b) f(x) = x + |x|, a = -1. c) f(x) = (x - 1)/(x^2 - 1), a = 1. In which cases do lim {x -> a-} f(x) and lim {x -> a} f(